COMP 175 | COMPUTER GRAPHICS
Lecture 12:
Recursive Ray Tracer2:
Advanced Techniques and Data Structures
COMP 175: Computer Graphics
April 7, 2016
Remco Chang
12 – Recursive Ray Tracer 2
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COMP 175 | COMPUTER GRAPHICS
Review

Ray Intersect (Assignment 4): questions / comments?

Review of Recursive Ray Tracer:

Two (or three) types of secondary rays

Lighting equation
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Blending and texture mapping
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The value of Epsilon…
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
Questions?
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
Glowing Ball

Now that you see how complicated (and slow) real rendering is,
how do you do some common techniques that appear “cool” but
run in real time?
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For example, a glowing floating ball?
https://www.youtube.com/watch?v=kAG1MbDeDps
Short answer: fake it with texture maps
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COMP 175 | COMPUTER GRAPHICS
Advanced Techniques (Shadows 1/2)
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Total hack: Project object on ground plane and render it as black
polygon. Eye less sensitive to exact geometry than to contrast of
shadow
Projective shadows (shadow mapping)
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Use orthographic or perspective matrix as CTM to project object (shadow caster)
against another object; render the shadow object as another primitive
Good for single light scenes with shadows cast
on planes
Shadow volumes
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Extrude silhouette volume from object
(shadow caster) to infinity in the
direction opposite the light source
Silhouette edge is made up of adjacent
polygons, any point within the volume
is in shadow
Implemented by former cs123 ta Kevin Egan and
former PhD student and book co-auther Prof.
Morgan McGuire, on nVidia chip
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
Advanced Techniques (Shadows 2/2)

Shadow Maps

Transform camera to each directional light source
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Render the scene from the light source POV, keeping the same far
clipping plane, update z-buffer only
Read the z-buffer back and apply as a texture (shadow map) to the
scene projected from the light’s POV
Render the scene from the eye point. At each pixel, if distance from
light is greater than in the shadow map, the object is in shadow.
Major aliasing problem, but implemented in hardware
Light
Shadow Map
Remco Chang
By keeping the same
far clipping plane,
relative distances in
Z are preserved
June
12 –21,
Recursive
2016 Ray Tracer 2
6
6/49
COMP 175 | COMPUTER GRAPHICS
Advanced Techniques (Environment Mapping)
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A way to render reflections (also called reflection mapping)
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Simulates reflections with a texture map applied to the enclosing
sphere or cube of an object
Texture map can either be an existing image or a rendering of the
scene from the object’s perspective
Shiny metal effect using environment map.
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
Cube environment map is created by stitching together 6 textures,
each generated by rendering the scene from the object’s POV
Ray from eye is reflected over surface normal and intersected with
the environment map to get texture coordinates.
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COMP 175 | COMPUTER GRAPHICS
Advanced Techniques (Surface Detail)

Overview: There are several
hacks to approximate the
lighting and shading of complex
geometry without actually
describing the geometry – only
used for fine detail
Perturb geometric normals with bump map
Desired Geometry
Approximate geometry at render time with
displacement map
Replace normals of simpler geometry
with normal map
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COMP 175 | COMPUTER GRAPHICS
Surface Detail (Bump Mapping)
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Texture mapping a rough surface onto an object doesn’t achieve desired
effect. Illumination is too uniform
Blinn’s technique: Use an array of values (stored in a texture called the
“bump map”) to perturb the surface normals by calculating gradient of
bump map and adding it to the surface normal
Evaluate lighting model with perturbed normals
Effect is convincing, but objects still look smooth at silhouette edges
Spherical Surface
Remco Chang
12 – Recursive Ray Tracer 2
nVidia
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COMP 175 | COMPUTER GRAPHICS
Bump Mapping Example

The tin foil on
this Hershey’s
Kiss looks too
smooth
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
Bump Mapping Example

Bump Mapped:
Tin foil look
with no
increase in
model
complexity.

Edges still don’t
look quite right
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COMP 175 | COMPUTER GRAPHICS
Normal Mapping
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Similar to bump mapping, but better for highly detailed
surfaces with drastically varying normals
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Level of detail is now limited by normal map resolution, not
number of triangles
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Bump mapping uses a single channel height field to perturb normals,
while normal mapping uses a three channel map to actually replace
existing surface normal
x, y, and z components of the desired normal, stored in the r, g, and b
channels of an image
Runtime surface normal determination is as easy as a texture lookup
Limitations:
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
Same as bump map, silhouette edges lack the added detail
Though easy to use at runtime, normal maps are difficult to generate
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COMP 175 | COMPUTER GRAPHICS
Normal Mapping Examples
Render showing actual,
simple underlying
geometry

When a normal map is
applied, there appears to
be much more detail in
the model than actually
is
Normal mapping can completely alter the perceived geometry of a
model
Image courtesy of www.anticz.com
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COMP 175 | COMPUTER GRAPHICS
Normal Map Creation (1/3)

While bump maps can be simply drawn, normal maps
hold meaningful data in three overlapping channels
Normal map for a
face. Red, green,
and blue channels
correspond to x, y,
and z normal
components
www.bitmanagement.com
Normal map for
brick street

www3.telus.net
Normal maps are not visually intuitive like height
maps. How are they created?
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COMP 175 | COMPUTER GRAPHICS
Normal Map Creation (2/3)

Normal maps must be generated by specialized software, such as
Pixologic’s Zbrush (www.pixologic.com)
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A high resolution model is created by an artist and its normals are used to
generate a normal map for a low resolution model.
3D geometry
Low-res mesh
(~5,000 polygons)
Normal Map
High-resolution model
(~2 million polygons)
Remco Chang
12 – Recursive Ray Tracer 2
Final render with normal map applied to low-res mesh
(and lighting and texture maps). This is only 5,000
triangles!
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COMP 175 | COMPUTER GRAPHICS
Normal Mapping Creation (3/3)

How it works:
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Artist creates a high resolution model with complex geometry (video)
High resolution model simplified using mesh decimation to a model with
much fewer triangles
Normal map generation:
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Let 𝑀 be the complex original mesh and 𝑆 the simplified mesh
To generate the normal to map to a point 𝑃𝑆 on 𝑆, find the
point 𝑃𝑀 on 𝑀 such that the Euclidean distance
between 𝑃𝑆 and 𝑃𝑀 is minimal.
𝑃𝑀 can be found by having each face of M stored in a grid and indexing
into this grid with coordinates collected during mesh simplification.
𝑆
We find the minimum perpendicular distance from 𝑃𝑆 to each
of 𝑀’s nearby facets
𝑃𝑆
Compute the normal at 𝑃𝑀 by averaging normals
at surrounding vertices. Place this value in
triangular normal map for the single face.
𝑃𝑀
Finally, stitch these triangular normal maps
together to create a normal map for the entire
simple surface.
𝑀
Note: The resolution at which
normals are sampled is user defined
See SIGGRAPH paper: “A general method for preserving attribute values
(again, time vs quality tradeoff)
on simplified meshes” by Cignoni, Montani, Rocchini, Scopigno
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COMP 175 | COMPUTER GRAPHICS
Other Surface Detail Techniques (1/3)
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Displacement Map:

The actual geometric position of points over the surface are displaced
along the surface normal according to the values in a texture
The displacement map used to render the
scene. The intensity value indicates the
height at a given point.
http://www.creativemac.com/2004/02_feb/tutorials/koc4ddisplace040224.htm
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
Other Surface Detail Techniques (2/3)
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Parallax Mapping:
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Displaces texture coordinates of a surface point by a function of the
view angle relative to the surface normal and the value on the height
map at that point
Steep Parallax Mapping:

Similar scheme to produce
self-occlusion self-shadows
of parallax mapped surface;
More efficient and reduces
swim effect for high
frequency maps (Morgan
McGuire Ph.D. ’06 and Max
McGuire)
Texture
Mapped
Remco Chang
12 – Recursive Ray Tracer 2
Texture and Parallax
Mapped
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COMP 175 | COMPUTER GRAPHICS
Other Surface Detail Techniques (3/3)
Texture Mapped
Normal Mapped: Increased
depth in rock wall
Remco Chang
Parallax Mapped: Increased
depth of whole texture but
unwanted “swim”
Steep Parallax Mapped: Very realistic depth in
effect. Textures look like they
whole picture. Lion head, columns cast self
should be 3D but they stay
shadows
embedded in surface,
unsettling to the eye when
moving
http://graphics.cs.brown.edu/games/SteepParallax/index.h
tml
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COMP 175 | COMPUTER GRAPHICS
Questions?
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
How to Speed Up My Ray Tracer?!
Remco Chang
12 – Recursive Ray Tracer 2
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COMP 175 | COMPUTER GRAPHICS
What Would You Do?

In the example below, we have to compute 8
intersection tests to find that we intersected with
nothing

Is there a way to speed this up?
Ray
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
Screen-Based Approach

We can project the objects onto the screen, and
compute the bounding boxes in 2D.


No need to cast rays where the screen is empty
If we use openGL to render the scene once, we can
(roughly) determine what object(s) we need to
intersect with!
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12 – Recursive Ray Tracer 2
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COMP 175 | COMPUTER GRAPHICS
What Would You Do?

In the example below, we have to compute 8
intersection tests to find the nearest object

Is there a way to speed this up?
Ray
Remco Chang
12 – Recursive Ray Tracer 2
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COMP 175 | COMPUTER GRAPHICS
Bounding Volumes

Use bounding volumes (that are easier/
cheaper to compute intersection tests
for)
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Spheres and boxes are most common
It’s typical to use axis-aligned bounding
boxes (makes math easier – no coordinate
transform)
Works very well for tight bounding boxes
Do this recursively to get a set of
hierarchical bounding boxes (bottom-up
or top-down)

Can be applied to the scene graph if the
objects in the scene are organized spatially
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COMP 175 | COMPUTER GRAPHICS
Creating a Bounding Volume Hierarchy (BVH)

Using a bottom-up approach
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COMP 175 | COMPUTER GRAPHICS
Bounding Volume Hierarchy (BVH) Example
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COMP 175 | COMPUTER GRAPHICS
Different Ways To Partition a Space
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Bounding volume hierarchy
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Grid
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Octree
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kd-Tree (k-dimensional tree)
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BSP-Tree (binary space partitioning trees)
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COMP 175 | COMPUTER GRAPHICS
Grids
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Divide the world into uniform cells
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Each cell stores the objects contained within it
Use line-scanning algorithm to traverse
Intersect with each cell along the way
0
1
2
3
4
5
6
0
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COMP 175 | COMPUTER GRAPHICS
Grids
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Problems:
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Choosing the right number of cells isn’t the most obvious
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Too many cells -> need to check a lot of each ray
Too few cells -> speed up can be reduced (too many objects in each one)
Some cells could be empty, while others have tons of objects within it

Note that this approach is not hierarchical, which means
that we don’t get the nice 𝑂(log 𝑛) benefit

Need to be careful in deciding what objects belong to a cell
(vertex, edge, face)
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But very useful for many reasons:
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Easy to implement in hardware
Good for an animated scene (an object moving to a different cell
doesn’t affect the other objects or other cells)
Remco Chang
12 – Recursive Ray Tracer 2
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COMP 175 | COMPUTER GRAPHICS
Octree
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Basically this is an adaptive
grids approach
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Generate more cells when
needed, but leave the white
spaces alone
Cells are still axis-aligned
Cells will no longer have the
same size
Remco Chang
12 – Recursive Ray Tracer 2
2-D example (called a quadtree)
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COMP 175 | COMPUTER GRAPHICS
Octree

Constructed using a top-down
approach
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Recursively split the world into 8
cells
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
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Terminates when the depth is too high,
or
Terminates when the number of
objects per cell falls below a threshold
Similar problem as grids, need to
be careful in determining which
objects belong to which cells
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COMP 175 | COMPUTER GRAPHICS
Octree
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
Start at the root node (cell), if it intersects, recursively check all 8
children (or 4 for quadtrees)
If a leaf node (cell) is empty, terminate. Otherwise, check all
objects within the cell
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COMP 175 | COMPUTER GRAPHICS
Octree
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
For each ray, the expected runtime is 𝑂(log 𝑛)
Problems:
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Updating an octree could be tricky (for an animated scene)
Finding neighbors of a cell isn’t trivial
For scenes that have objects distributed non-uniformly, the depth
could become very high
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COMP 175 | COMPUTER GRAPHICS
Kd-Tree
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Improves upon octrees in that it is a little
more “intelligent” in finding the split points
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Skips large swaths of empty spaces early
Zeros in on areas with high complexity (lots of
objects)
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In the worst case, a kd-Tree will be exactly
like a quadtree (for k=2), or an octree (for
k=3)

Like grids and octrees, kd-trees also use
axis-aligned bounding boxes (for checking
intersection is fast)

Unlike octrees, kd-Trees split in one
dimension at a time
Remco Chang
12 – Recursive Ray Tracer 2
Octree
kd-Tree
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COMP 175 | COMPUTER GRAPHICS
kd-Tree: Choosing a Split Plane
y=1
0
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
x=1
(Recursively) splitting a middle results in a quadtree
During traversal, cost of entering the right side is much
higher
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COMP 175 | COMPUTER GRAPHICS
kd-Tree: Choosing a Split Plane
y=1
0


x=1
Splitting at the medium balances out the cost, but
During traversal, the chance of entering the left hand
side is much higher (the area is larger)
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COMP 175 | COMPUTER GRAPHICS
kd-Tree: Choosing a Split Plane
y=1
0


x=1
Cost-optimized split: attempts to balance the cost of entering
and the probability of entering the two sides
Isolates the complex geometry quickly, creating large, (mostly)
empty spaces that can be efficiently computed and discarded
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COMP 175 | COMPUTER GRAPHICS
kd-Tree: Choosing a Split Plane
y=1
0


x=1
Further subdivision does not result in big gains (compared to
octrees)
But the point is that we isolated the complex geometry in one
split (which would have taken octree much longer to find)
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COMP 175 | COMPUTER GRAPHICS
kd-Tree: Caution with Edge Cases
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Notice the ray from left to right
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When it enters F, it will do an intersection test with the green rectangle
You will be tempted to say that you’re done and return the green rectangle as the
nearest object (because F is in front of G)
However, one can see that the clear nearest object is the yellow rectangle…
Solution, check the t-value against the bounding box t-value

Which will show that the t-value for the green rectangle is behind F.
Remco Chang
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COMP 175 | COMPUTER GRAPHICS
Cells
BVH
Octree
Remco Chang
kd-Tree
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COMP 175 | COMPUTER GRAPHICS
Binary Space Partition (BSP) Tree

Not really used for ray tracer, but for visible surface
determination
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This algorithm replaces the use of z-buffers
Fast to construct. Used in video games (like Quake 3)
Similar to kd-Tree, but NOT axis-aligned
BSP-0:
Initial Scene
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COMP 175 | COMPUTER GRAPHICS
BSP Trees
BSP-1: Choose any polygon (e.g., polygon 3) and subdivide others
by its plane, splitting polygons when necessary
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COMP 175 | COMPUTER GRAPHICS
BSP Trees
BSP-2: Process front sub-tree recursively
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COMP 175 | COMPUTER GRAPHICS
BSP Trees
BSP-3: Process back sub-tree recursively
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COMP 175 | COMPUTER GRAPHICS
BSP Trees
BSP-4: An alternative BSP tree with polygon 5 at the root
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COMP 175 | COMPUTER GRAPHICS
BSP-Trees

Remember that this is used for determining the
order in which objects need to be drawn on screen

So the order in which the objects are drawn will be
from back to front, but takes into account of the
direction of the face and where the eye is
(recursively)
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COMP 175 | COMPUTER GRAPHICS
BSP-Trees
void BSP_displayTree(BSP_tree* tree) {
location = tree->find_location(eye);
if ( location > 0 ) //eye in front of object
if ( viewer is in front of root ) {
BSP_displayTree(tree->backChild);
displayPolygon(tree->root);
BSP_displayTree(tree->frontChild);
}
else {
//eye behind the object
BSP_displayTree(tree->frontChild);
// ignore next line if back-face
// culling desired
displayPolygon(tree->root);
BSP_displayTree(tree->backChild);
}
}
}
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COMP 175 | COMPUTER GRAPHICS
Summary
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
Which data structure would be best to use in the following situations?
An animation where only the camera moves?


A fast-paced video game with lots of interactive, moving objects?


octree
Rendering scenes in a Pixar-like movie?


regular grid or BSP-Tree
A very balanced and even scene with objects uniformly distributed
throughout the whole scene?


kd-Tree
depends on the shot; most likely a combination of techniques are used
Your ray-tracing assignment if you are really fed up with your rendering
speed?

kd-Tree is best, but balance your decision with difficulty in implementing
the solution…
Remco Chang
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